Exam Practice Worksheet

Differentiation • SPM Paper 1 & 2

SPM KSSM Diagnostic Drill

Form 5 Chapter 2: Differentiation (Pembezaan)

Total Marks: 40 Marks • Time Allowed: 50 Minutes

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A Bahagian A: SPM Paper 1 Format [16 Marks]

Answer all questions
Soalan 1 [4 Marks]

Given that $y = \frac{1}{(2x - 3)^3}$.

(a) Find $\frac{dy}{dx}$. [2 marks]

(b) Hence, find the approximate change in $y$ when $x$ increases from $2$ to $2.02$. [2 marks]

Soalan 2 [4 Marks]

The curve $y = \frac{x^2 - 1}{x + 2}$ has a tangent parallel to the line $y = 5x - 7$. Find the $x$-coordinates of the points on the curve where this tangent occurs.

Soalan 3 (Paper 1 Sec B) [8 Marks]

A spherical weather research balloon is being inflated with helium gas at a constant rate of $120\pi\text{ cm}^3\text{s}^{-1}$. [Volume of sphere $V = \frac{4}{3}\pi r^3$, Surface area $A = 4\pi r^2$]

(a) Find the rate of change of the radius of the balloon when its radius is $10\text{ cm}$. [3 marks]

(b) Calculate the rate of change of the surface area of the balloon at that instant. [3 marks]

(c) Determine the percentage increase in the surface area of the balloon when the radius increases by $1.5\%$. [2 marks]

B Bahagian B: SPM Paper 2 Format & Real-World KBAT [24 Marks]

Answer all questions
Soalan 4 (Paper 2 Standard) [10 Marks]

A curve has the equation $y = 2x^3 - 9x^2 + 12x - 3$.

(a) Find the coordinates of the two turning points of the curve. [4 marks]

(b) Determine the nature of each turning point using the second derivative test. [3 marks]

(c) Find the equation of the tangent to the curve at the point of inflection. [3 marks]

Soalan 5 (KBAT Industrial Petroleum Tank Optimization) [8 Marks]
An engineering corporation designs a closed cylindrical petroleum storage tank of radius $r$ meters and height $h$ meters. The tank must hold a fixed capacity volume of $250\pi\text{ m}^3$. The heavy-duty reinforced steel for the top and bottom circular bases costs RM 80 per $\text{m}^2$, while the curved cylindrical side wall costs RM 50 per $\text{m}^2$.

(a) Show that the total construction cost $C$, in RM, is given by $C = 160\pi r^2 + \frac{25000\pi}{r}$. [3 marks]

(b) Determine the optimal radius $r$ and height $h$ that will minimize the total construction cost. [4 marks]

(c) Calculate this minimum construction cost, to the nearest Ringgit Malaysia. [1 mark]

Soalan 6 (Non-Routine Synthesis) [6 Marks]

Given that $y = 3x^2 - 5x + 4$.

(a) Find the coordinates of the point on the curve where the rate of change of $y$ is five times the rate of change of $x$. [3 marks]

(b) If $x$ changes by $p\%$, find the percentage change in $y$ at $x = 2$ in terms of $p$. [3 marks]